Difficulty: Intermediate | Prerequisites: Consumer theory basics (budget constraints, indifference curves, marginal utility). Familiarity with partial derivatives helps.
Competitive equilibrium sits at the intersection of consumer theory and market theory. You have been learning how individual consumers maximise utility given a budget constraint; now the question becomes: when many consumers trade in a market, what prices make everyone's plans consistent? This is the bridge from "one consumer choosing" to "a whole economy in balance." If you are comfortable with Cobb-Douglas utility and demand derivation, this topic clicks into place quickly. If not, revisit those first.
A competitive equilibrium is the set of prices at which every consumer maximises their utility and all markets clear (total demand equals total supply for every good). You find it by deriving each consumer's demand function, summing demands, setting the total equal to the total endowment, and solving for the unknown price.
Utility function
A mathematical representation of a consumer's preferences over bundles of goods. Think of it as a scoring rule: higher score means the consumer prefers that bundle.
Marginal utility (MU)
The additional utility gained from consuming one more unit of a good, holding everything else constant. In simple terms, it is the rate at which your "score" rises as you get more of one good.
Marginal rate of substitution (MRS)
The rate at which a consumer is willing to trade one good for another while staying equally happy. Formally, MRS = MUx / MUy. In simple terms, it answers: "How many units of Y would I give up for one more unit of X?"
Cobb-Douglas utility
A utility function of the form U(X,Y) = X^a * Y^b. It produces "nice" demand functions where the consumer spends a fixed fraction of income on each good. This is the workhorse functional form in intermediate micro.
Budget constraint (with endowments)
The set of affordable bundles when a consumer's income comes from selling their initial endowment at market prices. Income = Px * (endowment of X) + Py * (endowment of Y).
Demand function
The quantity of a good a consumer wants to buy, expressed as a function of prices and income. For Cobb-Douglas utility U = X^a * Y^b, the demand for X is: Dx = (a / (a + b)) * (M / Px).
Competitive equilibrium
A set of prices at which every consumer maximises utility subject to their budget constraint AND total demand equals total supply in every market. In simple terms, prices are "right" when everyone is happy with their choice and the market has no shortages or surpluses.
Market clearing condition
The requirement that total quantity demanded equals total quantity available (total endowment) for each good. This is the equation you solve to find the equilibrium price.
Numéraire
A good whose price is set to 1 as a reference point. Only relative prices matter in general equilibrium, so you can always normalise one price. In this problem, Py = 1 is the numéraire.
For Cobb-Douglas utility U(X,Y) = X^a * Y^b, the consumer spends fraction a/(a+b) of income on X and fraction b/(a+b) on Y.
Consumer A: U_A = X * Y (so a = 1, b = 1)
Demand for X: Dx_A = M_A / (2 * Px)
Demand for Y: Dy_A = M_A / (2 * Py)
Consumer B: U_B = X * Y^3 (so a = 1, b = 3)
Demand for X: Dx_B = M_B / (4 * Px)
Demand for Y: Dy_B = 3 * M_B / (4 * Py)
The exponents tell you spending shares. Consumer A splits income 50/50 across the two goods. Consumer B spends 1/4 of income on X and 3/4 on Y.
When consumers start with endowments rather than cash, income equals the market value of what they own.
Consumer A's endowment: X = 12, Y = 10, so M_A = 12 * Px + 10 * Py
Consumer B's endowment: X = 12, Y = 10, so M_B = 12 * Px + 10 * Py
With the numéraire Py = 1, both incomes simplify to M = 12Px + 10.
Total demand for X must equal total endowment of X (which is 12 + 12 = 24).
Dx_A + Dx_B = 24
M_A / (2Px) + M_B / (4Px) = 24
Substitute M_A = M_B = 12Px + 10:
(12Px + 10) / (2Px) + (12Px + 10) / (4Px) = 24
Multiply through by 4Px:
2(12Px + 10) + (12Px + 10) = 96Px
24Px + 20 + 12Px + 10 = 96Px
36Px + 30 = 96Px
30 = 60Px
Px = 0.5
Plug Px = 0.5 and Py = 1 back into the demand functions. Income for both consumers: M = 12(0.5) + 10 = 16.
Consumer A: Dx = 16 / (2 * 0.5) = 16 units of X, Dy = 16 / (2 * 1) = 8 units of Y
Consumer B: Dx = 16 / (4 * 0.5) = 8 units of X, Dy = 3(16) / (4 * 1) = 12 units of Y
Check: total X demanded = 16 + 8 = 24 (matches endowment). Total Y demanded = 8 + 12 = 20 (matches endowment). The market clears.
At equilibrium, each consumer's MRS equals the price ratio Px/Py.
MRS_A = MUx/MUy = Y/X = 8/16 = 1/2
Price ratio = Px/Py = 0.5/1 = 1/2
They match, confirming the equilibrium.
Cobb-Douglas demand (general form)
For U = X^a * Y^b:
D_X = \frac{a}{a+b} \cdot \frac{M}{P_X} \qquad D_Y = \frac{b}{a+b} \cdot \frac{M}{P_Y}Income from endowments
M_i = P_X \cdot \omega_{X,i} + P_Y \cdot \omega_{Y,i}where omega denotes the endowment of each good for consumer i.
Market clearing
\sum_i D_{X,i}(P_X, P_Y) = \sum_i \omega_{X,i}One equation per good, but by Walras' Law you only need to clear n - 1 markets.
MRS at equilibrium
MRS = \frac{MU_X}{MU_Y} = \frac{P_X}{P_Y}Competitive equilibrium is the theoretical backbone of how economists think about markets reaching balance. The same logic underpins models of international trade (countries "endowed" with different resources trading at world prices) and financial markets (investors with different portfolios trading securities until prices reflect aggregate preferences).
Students often think you need to clear every market separately. You do not. Walras' Law tells you that if n - 1 markets clear, the last one clears automatically. So with two goods, clearing the X market is enough.
Students sometimes forget that income is endogenous when consumers have endowments. Income depends on prices, so when Px changes, income changes too. You cannot treat M as a fixed number.
A common error is using the wrong spending share. For U = X * Y^3, the exponents are 1 and 3, giving shares of 1/4 and 3/4, not 1/2 and 1/2. Always add the exponents first.
Students occasionally set demand equal to zero or to one consumer's endowment instead of the total endowment across all consumers.
⚠️ Finding Px when Py = 1 is a staple exam question. Practise the full derivation until you can do it without notes.
⚠️ You will almost certainly be asked to verify equilibrium by checking that MRS = Px/Py. Always do this as a final step.
⚠️ The Cobb-Douglas spending-share shortcut (exponent / sum of exponents) saves enormous time. Know it cold.
⚠️ Expect variations: different utility functions, asymmetric endowments, or three goods. The method is identical; only the algebra changes.
True or False: In a two-good economy, if the market for X clears, the market for Y must also clear.
Fill in the blank: For U = X^2 * Y^3, the consumer spends ______ of income on X.
True or False: If Py is the numéraire, then Py = 0.
Fill in the blank: A consumer's income from endowments equals ______.
True or False: At competitive equilibrium, all consumers have the same MRS.
Answers: 1. True (Walras' Law). 2. 2/5. 3. False, Py = 1. 4. Px times endowment of X plus Py times endowment of Y. 5. True (all equal to the price ratio).
Q: Two consumers have Cobb-Douglas utility. Consumer A has U = X * Y and Consumer B has U = X * Y^3. Both are endowed with (12, 10). With Py = 1, what is the competitive equilibrium price of X?
A: Px = 0.5. Derive each consumer's demand for X using spending shares (1/2 for A, 1/4 for B), set the sum equal to total endowment of 24, substitute M = 12Px + 10, and solve.
Q: In the equilibrium above, how many units of Y does Consumer B demand?
A: 12 units. Dy_B = 3M / (4Py) = 3(16) / 4 = 12.
Q: How do you verify that the allocation is a competitive equilibrium?
A: Check two things. First, total demand equals total supply for each good. Second, each consumer's MRS equals the price ratio Px/Py.
Q: Consumer C has utility U = X^2 * Y and endowment (6, 8). With Py = 1, write down C's demand for X as a function of Px.
A: Spending share on X = 2/3. Income = 6Px + 8. So Dx_C = (2/3)(6Px + 8) / Px = (4Px + 16/3) / Px = 4 + 16/(3Px).
This connects directly to the Edgeworth Box, which visualises the same two-consumer, two-good economy. The competitive equilibrium corresponds to a point on the contract curve inside the box.
Walras' Law, used here to reduce the number of market-clearing equations, reappears in general equilibrium theory with many goods and many consumers.
The MRS condition at equilibrium (all consumers' MRS equal the price ratio) is the same first-order condition that defines Pareto efficiency. This is the content of the First Welfare Theorem: competitive equilibria are Pareto efficient.
Competitive equilibrium, general equilibrium, Walrasian equilibrium, Cobb-Douglas utility, Cobb-Douglas demand, marginal rate of substitution, MRS, market clearing, endowment economy, exchange economy, numéraire, price ratio, spending shares, budget constraint with endowments, Walras' Law, First Welfare Theorem, Pareto efficiency, ECON 302, intermediate microeconomics, Penn State